Scilab Function
Last update : 12/1/2004

findx0BD - Estimates state and B and D matrices of a discrete-time linear system

### Calling Sequence

  `[X0,B,D] = findx0BD(A,C,Y,U,WITHX0,WITHD,TOL,PRINTW)`
  `[x0,B,D,V,rcnd] = findx0BD(A,C,Y,U)`

### Parameters

- `A`: state matrix of the system
- `C`: C matrix of the system
- `Y`: system output
- `U`: system input
- `WITHX0`: a switch for estimating the initial state x0.
  - `=` 1: estimate x0;
  - `=` 0: do not estimate x0.

  Default: WITHX0 = 1.

- `WITHD`: a switch for estimating the matrix D.
  - `=` 1: estimate the matrix D;
  - `=` 0: do not estimate the matrix D.

  Default: WITHD = 1.

- `TOL`: the tolerance used for estimating the rank of matrices. If TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition number. Default: prod(size(matrix))*epsilon_machine where epsilon_machine is the relative machine precision.
- `PRINTW`: a switch for printing the warning messages.
  - `=` 1: print warning messages;
  - `=` 0: do not print warning messages.

  Default: PRINTW = 0.

- `X0`: intial state of the estimated linear system.
- `B`: B matrix of the estimated linear system.
- `D`: D matrix of the estimated linear system.
- `V`: orthogonal matrix which reduces the system state matrix A to a real Schur form
- `rcnd`: estimates of the reciprocal condition numbers of the matrices involved in rank decisions.

### Description

findx0BD Estimates the initial state and/or the matrices B and D of a discrete-time linear system, given the (estimated) system matrices A, C, and a set of input/output data.

[X0,B,D] = findx0BD(A,C,Y,U,WITHX0,WITHD,TOL,PRINTW) estimates the initial state X0 and the matrices B and D of a discrete-time system using the system matrices A, C, output data Y and the input data U. The model structure is :

         x(k+1) = Ax(k) + Bu(k),   k >= 1,
         y(k)   = Cx(k) + Du(k),

The vectors y(k) and u(k) are transposes of the k-th rows of Y and U, respectively.

[x0,B,D,V,rcnd] = findx0BD(A,C,Y,U) also returns the orthogonal matrix V which reduces the system state matrix A to a real Schur form, as well as some estimates of the reciprocal condition numbers of the matrices involved in rank decisions.

        B = findx0BD(A,C,Y,U,0,0)  returns B only, and
    [B,D] = findx0BD(A,C,Y,U,0)    returns B and D only.

### Examples

    //generate data from a given linear system
    A = [ 0.5, 0.1,-0.1, 0.2;
          0.1, 0,  -0.1,-0.1;
         -0.4,-0.6,-0.7,-0.1;
          0.8, 0,  -0.6,-0.6];
    B = [0.8;0.1;1;-1];
    C = [1 2 -1 0];
    SYS=syslin(0.1,A,B,C);
    nsmp=100;
    U=prbs_a(nsmp,nsmp/5);
    Y=(flts(U,SYS)+0.3*rand(1,nsmp,'normal'));

    // Compute R
    S=15;L=1;
    [R,N,SVAL] = findR(S,Y',U');

    N=3;
    METH=3;TOL=-1;
    [A,C] = findAC(S,N,L,R,METH,TOL);

    [X0,B,D,V,rcnd] = findx0BD(A,C,Y',U');
    SYS1=syslin(1,A,B,C,D,X0);

    Y1=flts(U,SYS1);
    xbasc();plot2d((1:nsmp)',[Y',Y1'])

### See Also

`findBD`, `inistate`,